Should HQIC be high or low?

Lower is preferred among models that are validly comparable under the same convention.

Is HQIC the same as AIC or BIC?

No. They use different complexity penalties and can therefore rank models differently.

Can HQIC replace an independent forecast evaluation?

No. It is a model-selection criterion, while an appropriate forecast evaluation measures performance on predictions made under realistic conditions.

AI Glossary / AI and machine learning / Intermediate

Hannan - Quinn information criterion

Definition

The Hannan - Quinn information criterion is a score for comparing statistical models that balances goodness of fit against model complexity. Commonly abbreviated HQIC or HQC, it favors the candidate with the lower score when models are compared on a compatible basis.

What it means

Adding parameters often improves a model's fit to the data used to estimate it. That improvement may represent useful structure, but it can also reflect unnecessary flexibility. An information criterion adds a complexity penalty to discourage selecting a larger model solely because it fits the observed sample better.

In a common likelihood-based form, HQIC equals negative twice the maximized log likelihood plus 2k times the natural logarithm of the natural logarithm of n. Here k is the counted number of estimated parameters and n is the relevant sample size.

The double logarithm gives HQIC a sample-size-dependent penalty different from AIC and BIC. The exact parameter count and likelihood convention must be checked in the statistical setting and software being used.

HQIC is a relative comparison tool. A low value does not prove that a model is correct, that its forecasts will be accurate, or that its assumptions fit the application. It helps choose among the candidates actually considered.

How it works

Specify candidate models that can be compared meaningfully. Fit them to the same response data with compatible likelihood definitions and observation counts. Time-series lag choices can change usable observations unless the comparison is designed carefully.

Obtain each model's maximized log likelihood and the appropriate number of parameters. Include relevant estimated terms according to the chosen convention rather than counting only the predictors visible in a formula.

Compute the fit term and complexity penalty, then add them. A better likelihood lowers the fit contribution, while additional parameters increase the penalty. The preferred score reflects their balance.

Use the ranking alongside diagnostics and an evaluation strategy suited to the task. For forecasting, that may include validation respecting time order. Check residual behavior and practical interpretability instead of treating one numerical criterion as the entire model review.

Three practical examples

1. A forecasting analyst compares several autoregressive orders. HQIC provides one way to balance additional lag parameters against improved likelihood on a common estimation sample.

2. A statistics team compares nested likelihood-based models for a continuous outcome. It documents the parameter-count convention before interpreting differences between reported scores.

3. A training exercise compares AIC, BIC and HQIC rankings. Learners investigate why different complexity penalties can select different candidates from the same fitted models.

Think of it this way

Think of judging explanations by both how well they account for the evidence and how many adjustable details they require. Extra detail must earn its place. The analogy is limited because HQIC uses a specific mathematical penalty, not a subjective preference for short explanations, and it cannot assess every dimension of a model's usefulness.

Worked example

Suppose two models use the same 100 observations. Model A has maximized log likelihood -120 and 3 parameters. Model B has log likelihood -117 and 5 parameters.

The natural logarithm of the natural logarithm of 100 is approximately 1.527. Model A therefore has HQIC approximately 240 plus 2 times 3 times 1.527, or 249.16. Model B has approximately 234 plus 2 times 5 times 1.527, or 249.27.

Under this stated convention, A has the slightly lower score despite B's better fit. The difference is small, so the example should not be read as decisive practical superiority. It illustrates how a modest fit improvement can be offset by the added complexity penalty.

Illustrative case study

Fictional case: Brookside Planning compares demand models with different lag orders. An analyst initially copies HQIC values from separate software runs without checking their estimation periods.

She discovers that the longer-lag model used fewer observations. The team refits candidates on a consistent comparison window and documents how parameters are counted. It then reviews the criterion ranking alongside time-ordered forecast evaluation.

The selected model remains subject to monitoring as demand changes. This invented case demonstrates comparability and validation, not a claim that HQIC always chooses the best forecasting model.

Common misconceptions and mistakes

Mistake: Comparing scores from incompatible likelihoods or different response datasets. The numerical values may not support the intended ranking.

Mistake: Counting only regression slopes when the convention includes additional estimated parameters. Incorrect counts change the penalty.

Mistake: Treating the lowest score as proof of truth. HQIC selects relatively among candidates and does not independently validate their assumptions.

Business use

HQIC can contribute to a disciplined model-selection process, especially in statistical time-series work. Its value is greatest when candidate definitions, data windows and calculation conventions are transparent.

Business decisions should still consider forecast errors, operational costs and robustness to changed conditions. If two models have nearly identical scores, the simpler explanation or more stable behavior may be relevant, but those considerations should be stated rather than attributed to the criterion itself.

Check your understanding

A more complex model has a better log likelihood. Must it have a lower HQIC?

Answer: No. The additional parameter penalty may exceed the improvement in the likelihood term.

Related terms

Learn next

Learn log likelihood and AIC before comparing information criteria. Calculate all three scores for two invented candidates using the same observations, and identify exactly which penalty causes their rankings to differ.

Log likelihood

Sources

Stata, ARIMA model selection methods and formulas: https://www.stata.com/manuals/tsarimasoc.pdf. Official formula documentation consulted. Statsmodels HQIC reference: https://www.statsmodels.org/dev/generated/statsmodels.tools.eval_measures.hqic.html. Worked arithmetic and fictional case are original.

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